That’s a paper model of a polyhedral version of the “Boy’s surface” — the key half-way point in a singularity-free eversion of a sphere. (Eversion simply means “turning inside out,” like you would a sock or a t-shirt. Of course, a sphere has to intersect itself to be turned inside out in three dimensions.) And it was created by Konrad Polthier, who wrote an article about it an lots of other folding things. Here are a couple of examples:

Your challenge, should you choose to accept it, is to use the information in the article to construct an actual physical model of one of the more interesting shapes or surfaces featured there, and send a photo to [email protected] — if enough photos come that way, there will be a follow-up posting with some of the best. Good luck! (Some of those models are remarkably intricate — take a look and see.)